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Geometry Level 5

\[\large \dfrac{2\sqrt{19}}{3} \cos \left(\dfrac{1}{3} \arccos \left(\dfrac{7}{\sqrt{76}} \right) \right) - \dfrac{1}{3}\]

The expression above can be simplified into the form

\[\large a \left(\cos \left(\dfrac{b \pi}{e} \right) +\cos \left(\dfrac{c \pi}{e} \right)+\cos \left(\dfrac{d \pi}{e} \right) \right)\]

where \(a,b,c,d\) and \(e\) are positive integers with \( \gcd(b,e) = \gcd(c,e) = \gcd(d,e) = 1 \).

If all the angles mentioned above lie in the interval \( [ 0, \pi ] \), find the value of \(a+b+c+d+e\).

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